In a simple cosmology, the earth is supported by a turtle; that turtle is
supported by another; and so on "all the way down." The most prudent
reaction is that this is a charming piece of folklore not to be examined
too closely, either historically or physically. Here I will cast prudence
aside and ask whether it could make sense physically for the world to be
supported by an infinite tower of turtles.

The question started as a moment of musing. Does this infinite tower work
if we imagine it in the simplest physical setting: a homogeneous,
Newtonian gravitational field. Will the tower stand or fall? That is, will
the tower in its infinite entirely just fall downward in the gravitational
field, just as would any unsupported object? Or does the infinity of the
tower protect it from this fate, so that it can serve to hold up the
world? We seemed to have a well-posed question in Newtonian gravitation
theory. We would expect a simple and quick answer.

The moment of musing over this puzzle extended into moments as I found
that things are not quite so simple. There are apparently quite cogent
arguments for both possibilities, stand or fall. It took me a while to see
past the problems. I found the exercise quite entertaining. This piece is
written for anyone who might like a brief diversion into an innocent
puzzle that proves a little messier than you may expect.

2. A Preliminary Kinematical Puzzle

The chief concern here will be whether an infinite tower stands or falls
when we take proper account of the Newtonian forces at play. As a
preliminary, independently of any issue of physical forces, we might
expect that the tower of turtles cannot fall for a reason of kinematics:

There’s nothing for the tower to fall into!

We might reason as follows. For a body to fall, there must be empty space
under it, into which it can fall. But there is no empty space under the
infinite tower of turtles. So there is nowhere for it to fall. Thus it
cannot fall.

Another way to put the problem is that, for the tower to fall, it can
only fall into its own space. But its own space is already fully occupied
by turtles.

The solution to this puzzle will be familiar to anyone who has seen the Hilbert
Hotel. Having no place into which it can fall would be a problem for
a body of finite volume. The turtle tower, however, has infinite volume.
So it can fall in such a way that, during the fall, it occupies only a
part of the volume of space initially occupied.

We can see this using a “Hilbert hotel” construction. In the first unit
of time of fall:

This solves the kinematical problem. An infinite tower of turtles can
fall within the space it occupies.

3. The Puzzle: Arguments For and Against

If we take into account the forces acting on the turtles and the forces
between them, will the tower of turtles stand or fall? There are arguments
for and against; and they seem equally cogent.

3.1 The Balanced Force Argument For Standing

Consider any one turtle in the tower. The weight of the turtles above
presses down on it and, in addition, a gravitational force pulls it
downward. However it sits on the back of another turtle and that turtle
exerts a reaction force on the first turtle that exactly cancels these
downward forces.

It is like a brick in a wall. The brick carries the weight of the bricks
above it, as well as the force of gravity acting on it. Those forces are
balanced exactly by the reaction force from the brick beneath it.

Thus the turtle under consideration stands and does not fall. This
turtle, however, can be any of the infinitely many turtles in the tower.
That means that all the turtles in the tower stand. Since the behavior of
the tower as a whole is just the sum of the behaviors of all the
individual turtles in the tower, the tower stands.

If this argument seems too hasty, note how it fails for a finite tower of
turtles. The reasoning above holds for every turtle except the lowest. If
that turtle is unsupported, then there is no reaction force holding it and
it must fall. It is, as a result, unable to exert a reaction force on
the turtle above it, which also must fall. We proceed up the tower
to conclude that all the turtles and the world must fall.

The balanced forces argument depends essentially on their being an
infinity of turtles, so that there is no, lowest unsupported turtle. Every
turtle is supported by one beneath it.

3.2 The Unbalanced Force Argument for Falling

Consider the tower as a single huge object. It sits in a gravitational
field that exerts an external downward force on it. There is no
compensating external force to counteract it. Hence the tower must fall.

A slightly more sophisticated version applies Newton's

Force = mass x acceleration

(1) F = ma

There is a non-zero, net downward force acting on the tower. Through F =
ma, the tower is accelerated downward.

A complication gives us a temporary respite. If the tower consists of
infinitely many, equally massive turtles then the mass m in F=ma
is infinite. Since the gravitational force is proportional to the mass,
the gravitational force is infinite as well. Thus F=ma becomes:

Infinite force = infinite mass x a

This equation leaves the acceleration a undetermined. It could
be any value: zero, finite or infinite.

That the respite is temporary follows if we consider a fanciful
elaboration of the tower of turtles. What if masses m1, m2,
m3, ... diminish so that their sum is finite? Call
that sum Mt

where the constant g is the acceleration imparted by the
homogeneous gravitational field. Solving for a we have that the
tower accelerates downward with acceleration g. That is, a
= g.

3.3 The Limit Argument for Falling

A standard method for recovering the behavior of an infinite system is to
consider the behavior of finite systems and then take a limit as the
number of components becomes infinitely large.

To this end, consider a tower of finitely many -- N -- turtles.
For any value of N, we have a tower unsupported in the
gravitational field. It will fall with acceleration g.

If we now take the limit as N goes to infinity, we arrive at an
infinite tower falling with acceleration g.

3.4 The Limit Argument for Standing

There is another way to approach the this same limit of an infinite
tower. Consider a tower of finitely many -- N -- turtles. This
time, the lowest, Nth turtle is supported by a powerful rocket
motor. Its thrust is enough to hold up the entire weight of the tower. The
finite tower stands.

Now take the limit as the number of turtles N goes to infinity.
We once again end up with an infinite tower, but this time the tower
stands.

What of the rocket motor? It is tempting to imagine that the tower is
held up by the rocket motor, which now lives on "at infinity." But the
place "at infinity" is a fiction. There is no such place. The rocket motor
supports the lowest turtle in the tower. In the tower of the infinite
limit, there is no lowest turtle. Every turtle in the tower can be
assigned a number according to its position in the tower. Pick any number
N? Is that the turtle supported directly by the rocket motor? No!
There is another turtle, number N+1, beneath it.

In the limit, there is no rocket motor. There is just an infinite tower
of turtles that stands.

Now What?

We have a classic paradox. We have pairs of arguments, each apparently
quite cogent, but arriving at conclusions that contradict. Two conclude
that the tower falls. Two conclude that the tower stands.

The fun of the paradox is seeing through the logic of the arguments. Some
or possibly all of the arguments are flawed. Which are they?

Here I urge the reader to take a short break and ponder. I am giving my
solution below. But isn't it more fun to figure out your own solutions
first?

4. Solutions

The two limit arguments can be dispatched quickly. The false presumption
is that either limit process is a reliable way of inferring the behavior
of an infinite tower. That is already evident from the mere fact that we
have two ways of approaching the same limit of an infinite tower of
turtles, but the two ways ascribe different properties to the same
infinite tower.

The use of limiting procedures to infer the properties of infinite
systems is delicate and sometimes fraught. Their use can give
spurious results. For some further discussion, see my "Approximation
and Idealization..."

More subtlety is needed to resolve the contradiction produced by the two
arguments pertaining the the balance or imbalance of forces. The
resolution depends on a disanalogy between the case of a finite tower and
an infinite tower.

In the case of a finite tower, we determine its future behavior by
stipulating certain of its properties at present: the positions of the
objects and their initial velocities. Those same properties are not
sufficient to determine the future behavior of the infinite tower. Whether
it stands or falls requires specification of further properties: the
inter-object forces at the initial moment of time.

Specifying these forces one way gives a tower that falls, as the
unbalanced forces argument predicts. Specifying them another way gives a
tower that stands, as the balanced forces argument predicts.

In short, both balanced and unbalanced forces arguments succeed, but only
because they apply to different cases. Further specification of the
initial conditions allow to decide which case we have at hand.

Here is a more sustained development of this resolution.

4.1 Initial conditions for a finite tower

The initial conditions required for a finite tower are

(i) the positions of all the component turtles and
world;
(ii) their initial velocities (which are all zero).

In one case, the components include an immovable base at rest. What
results is a finite tower of turtles, resting on the base. It remains so
and does not fall.

In another case, the components do not include the immovable base. The
lowest turtle is unsupported and falls. With it, all the remaining turtles
fall.

4.2 Initial conditions for an infinite tower

The initial conditions required for an infinite tower are

(i) the positions of all the component turtles and world;
(ii) their initial velocities (which are all zero);
(iii) the initial inter-turtle and turtle-world forces.

The extra condition (iii) requires specification of the reaction forces
exerted between neighboring turtles and on the world.

Their specification is not needed in the finite case. Whether reaction
forces are there or not follows, for the finite tower, from the initial
conditions (i) and (ii).

If there is an immovable base, gravity cannot accelerate the lowest
turtle through the base, so the lowest turtle exerts a force on it.
Because of its immovability the base exerts an equal but opposite reaction
force on the lowest turtle. This lowest turtle is immobile. This reasoning
propagates up the tower. Each turtle exerts a loading force on the one
beneath. That turtle exerts an equal but opposite reaction force on the
one above. All turtles of the tower are immobile.

If there is no immovable base, the lowest turtle enters into free fall
and exerts no reaction force on the turtle above it. This next turtle
enters into free fall; and so on up the tower.

However, for the case of an infinite tower, initial condition (i) and
(ii) alone are insufficient to determine the future behavior of the tower.
We must posit in (iii) whether there are loading and reaction
forces between the turtles. If we posit their existence, then the initial
conditions lead to a tower that stands. If we posit none, we have a tower
that falls.

5. Reaction forces made explicit

Since the reaction forces can grow arbitrarily
large, it is simplest to conceive of springs that do not obey Hooke's law,
but instead a non-linear law, such that the force grows rapidly with the
compression of the spring.

A further complication is that systems consisting of an infinite chain of
mass-spring-mass-spring-... are generically indeterministic. The system
can spontaneously excite through disturbances that, loosely speaking,
propagate in from infinity. See Approximation
and Idealization... (Appendix). We can preclude them with a fourth
condition:

(iv) The positions of the turtles relative to each other remain fixed over
time.

A way see how this last conclusion works, is to make the presence or
absence of loading and reaction forces in the infinite tower visible by
connecting the turtles by springs.

A pair of loading and reaction forces between one turtle and the one
above will manifest as a compression of the spring connecting them. If
there is no force, then the spring is unextended and uncompressed. It is
at its equilibrium length in which in exerts no force. If there is a
force, then the spring is compressed.

How we set the initial condition in (iii) will then determine whether the
infinite tower of turtles will stand or fall:

• If we initialize the infinite tower so that each turtle is supported by
a reaction force from the turtle below it, the tower will stand. (Shown on
the right.) Those supporting reaction forces will persist unchanged over
time.

• If we initialize the infinite tower so that there are no reactions
forces supporting each turtle, the tower will fall. (Shown on the left.)
While it falls, the connecting springs will remain in their zero-force
uncompressed state. The tower will continue its free fall.

5.1 The governing equations

We can give a more substantive analysis in Newtonian physics of these two
cases.

Sign conventions: the x axis increases
downward in the direction of the field. Positive forces are directed
downward.

Let the mass of the world be m0 and the masses of
the turtles m1, m2, m3,
... as before. The world and turtles are located at positions x0,
x1, x2, x3,
... The net force acting on the ith object is

(4) f0 = m0a0= m0g + f01
fi
= miai = mig + fi,i-1
+ fi,i+1for i>0

where fik is the force exerted on the ith
object by the kth object. (The only cases we will consider are
those in which i and k are one number separated, so
that k=i-1 or k=i+1.) The
acceleration of each object is ai.

The weight of all the objects above the ith object is
transmitted to it as fi,i-1 which is the force
with which the i-1th object bears down on the ith
object. The ith object exerts a reaction force back on the i-1th
object of fi-1,i

Newton's third law requires that the force with which the ith
object acts on the kth object is equal but opposite in sign to
the force with which the kth object acts on the ith
object. That is:

(5) fik = -fki

5.2 The tower stands

The case of the tower standing arises when all the forces vanish. Then
the accelerations must also vanish:

(6) 0 = f0 = f1
= f2 = f3 =
... 0 = a0 = a1
= a2 = a3 = ...

We can find the forces fik that realize this case by
solving equations (4), (5) and (6) iteratively:

If the tower is initialized with these inter-object forces, the springs
will all be compressed by just the amount needed to sustain these forces.
Since the net force on each object is zero, the objects do not accelerate.
They remain separated by the same distances, so that the springs retain
their compression and the forces of (7) remain unchanged through time.

The tower stands, supported by the forces in (7).

5.3 The tower falls

The case of the tower falling arises when there are no inter-object
forces. That is, the springs connecting the objects are in their
uncompressed states. Then we have

(8) fik = -fki= 0

Equations (4) are then easily solved to give

(9) ai = g

All the objects fall with the same acceleration g. It follows
that the distance between them remains the same, so no inter-object forces
arise during the fall.

6. A center of mass theorem

The unbalanced force argument of Section 3.2 above assumed that what
determines the overall motion of the tower of turtles is the net external
force acting on the tower. A non-zero net external force, as is the force
of gravity, must accelerate it.

This assumption cannot be made without recalling the conditions that lead
to it. It is derived as a theorem in elementary mechanics from the
summation of all the forces acting on the component objects of the system.
The theorem provides a summary description of the combination of all those
forces. Under ordinary conditions, the summation leads to the familiar
result that we can ignore all the inter-object forces, when we consider
the system as a whole. The effect of all external forces is to impart an
acceleration to the system's center of mass that is proportional to its
total mass.

The theorem is easy to derive. Here is the derivation for the special
case of the tower of objects, that is, the world plus its supporting
turtles.

6.1 A center of mass theorem for a finite tower

For a finite tower with N turtles, equations of motion (4) must
be replaced by

These equations for the motion of each individual object then determine
an equation governing the center of mass of the whole tower. To see it,
sum the equations of (10) to find the total, net force F acting:

This vanishing is as expected. It tells us that the inter-obect forces
contribute nothing to the net force acting on the tower as a whole.

Combining, we recover the final expression

(12) F = M (d2/dt2)
X = Mg

That is, the net force F acting on the tower is just the
external gravitational force Mg and its effect is an
acceleration of the center of mass A = (d2/dt2)
X that conforms with F = MA.

6.2 A center of mass theorem for an infinite tower

Equation (12) shows us that we can ignore the inter-object forces in a
finite tower. We can treat the tower as a single object that is acted on
only by external gravitational forces and whose center of mass is
accelerated accordingly.

The unbalanced forces argument of Section 3.2 above assumed that we can
treat the infinite tower in the same way. For that to be so, we must have
a center of mass theorem for an infinite tower analogous to the one just
demonstrated for a finite tower.

As we shall now see, it turns out that we do not have a corresponding
center of mass theorem that vindicates the unbalanced forces argument. We
will see that the applicable theorem can sustain a tower that falls or one
that stands, according to the assumptions we make about the inter-object
forces.

To find the corresponding theorem, we proceed as in Section 6.1 by
summing all the forces in (4) acting on the individual objects in the
tower. We find:

However if this last mass M is infinite and the summation of
terms in mixi diverges, then the center of
mass X will be undefined. In this generic case, a center of mass
theorem cannot be recovered. The reasoning of the unbalanced forces
argument of Section 3.2 is blocked.

However, we can take the expedient of Section 3.2 and assume that the
total mass M is finite. In addition we can assume that the
positions xi do not grow too fast with i so
that the sum m0x0+ m1x1+ m2x2+
... converges. Then the center of mass X is defined and a center
of mass theorem is recoverable. (See here
for computation of specific cases.)

With these assumptions, the analysis proceeds as in Section 6.1 and we
arrive at

The crucial different is the presence of one additional term in the
summation of inter-object forces, which is indicated in enlarged text: +
fN,N+1). This is the
reaction force acting on the Nth object from the (N+1)th
object. This term does not appear in the analysis of the finite tower,
since there is no N+1th turtle.

If we proceed as before and regroup the terms in the summation of
intermolecular forces, we recover for this sum:

Might one object to the use of this infinite limit,
recalling how limits can lead to unwanted problems? If one rejects this
limit, then the net force on the entirety of the infinite tower is
undefined. That means that we cannot set up a center of mass theorem in
the first place and we must set aside concerns associated with it.

To conclude, we define the total force F on the tower
to be the limit:

F = Lim
(N → ∞)FN = 0

Since it vanishes then so must

(d2/dt2)
X = 0.

The center of mass X is unaccelerated.

That is, when the inter-object forces are given by (7) for a standing
tower, the applicable version of the center of mass theorem tells us that
there is no net force on the total system and its center of mass X
remains unaccelerated.

This result comes about because of a failure of a familiar intuition
concerning inter-object forces. We naturally assume that they all cancel
out and leave no residual force to act on the tower as a whole. That
assumption fails because of the infinity of the number of objects. For any
finite part of the tower, there is always a residual force, the reaction
force exerted on the Nth object by the N+1th
object, fN,N+1 = -MNg
. This residual force survives in the limit as the number of turtles
becomes infinite. It provides exactly the force -Mg needed to
counteract the external forces +Mg of gravity.

7. Conclusion

What this analysis concludes is that the usual presentation of the
problem of the infinite tower of turtles leaves the state of the tower
incompletely described. This incompleteness is what enables there to be
apparently cogent arguments for contradictory conclusions. If we want to
know whether the tower will stand or fall, we need more information than
the initial positions of the world and each turtle and their initial
velocities. We need also to know what forces are acting pairwise between
the world and all the turtles. One specification leads to a tower that
falls. Another specification leads to a tower that stands.

The longest and technically messiest part of the analysis is in Section
6. It deals with the argument that the tower must fall since, as a
totality, it is a body acted on by a gravitational force, but without any
external force to counteract it.

This unbalanced forces argument depends on the assumption that one can
neglect all the inter-object forces when one analyses the tower as a
whole. That assumption is supported by a center of mass theorem formulated
to apply to finite towers only.

We can develop a version of the center of mass theorem that applies to
infinite towers. Its scope is limited since, in generic cases, its
quantities diverge so no result is derivable. If we contrive circumstances
in which these divergences are avoided, then the infinity of turtles leads
to a curious result. The totality of forces acting between the towers do
not cancel. There is a residual that balances the external gravitational
forces exactly and allows the tower to stand.

One should not become too comfortable with this last result. If we allow
the sizes of the object to shrink so that the entirety fits into a finite
volume of space, this same mechanism will allow an infinity of objects to
sustain themselves in mid-air by (metaphorically) "pulling on their own
bootstraps." It also provides the architectural design for the foundations
of a castle that can float in the air.